Two proxy advisers recommended that Exxon Mobil Corp. and Chevron Corp. shareholders vote in favor of splitting the chair and CEO roles, creating a potential flash point with the boards at the oil giants’ annual meetings on May 27.

Glass Lewis & Co. and Egan-Jones Proxy Services say investors should vote against the companies and embrace proposals that would create independent chairs once the incumbents — Darren Woods at Exxon and Mike Wirth at Chevron — step down. Both advisory firms say a CEO who also leads the board raises a potential conflict of interest.

“An independent chair is better able to oversee the executives of a company and set a pro-shareholder agenda without the management conflicts that a CEO or other executive insiders often face,” Glass Lewis said in its report on Exxon.

Last year, almost 41% of Exxon investors voted in favor of the proposal, which would amount to one of the biggest corporate governance changes in the oil explorer’s history if passed.

Legal & General Investment Management, the U.K.’s largest asset manager and a top 20 Exxon shareholder, said it will vote in favor of the proposal, and it also plans to vote against Woods’s reelection over what it called a lack of ambition over climate change. The California Public Employees’ Retirement System, America’s largest state public pension fund, and the New York State Common Retirement Fund have also said they support splitting the chair and CEO roles.

Institutional Shareholder Services Inc., another big proxy adviser, said investors should reject the CEO-chair split at Exxon and Chevron, calling the powers of lead directors at both companies “robust.”

Exxon recently beefed up the responsibilities of its lead director to address some shareholder concerns. In its proxy filing, Exxon said having the CEO chair the board “ensures items of greatest importance for the business are brought to the attention of, and reviewed by, the board on a timely basis.”

In another filing on Wednesday, the company rejected Glass Lewis’s stance on the separation of the roles, pointing out that 90% of its board is independent and that only about 34% of S&P 500 companies have an independent chair.

Chevron’s board “believes that stockholder interests are best served when directors have the flexibility to determine the best person to serve as chairman, recognizing that no single leadership model is appropriate in all circumstances,” the company said in a statement.